A211034 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 1 (mod 3).

0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744, 10955, 15000, 18100, 23988, 31104, 36432, 46179, 57624, 66052, 81056, 98304, 110848, 132723, 157464, 175284, 205860, 240000, 264400, 305723, 351384, 383812, 438144, 497664, 539712, 609531

Offset: 0

Author

Clark Kimberling, Mar 30 2012

Keywords

• nonn
• easy

Comments

Also, the number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 2 (mod 3). A211033(n) + 2*A211034(n)=n^4 for n>0. For a guide to related sequences, see A210000.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,-6,6,0,4,-4,0,-1,1).

Crossrefs

Cf. A210000, A211033.

Programs

Mathematica

t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
u[n_] := u[n] = Sum[c[n, 3 k], {k, -2*n^2, 2*n^2}]
v[n_] := v[n] = Sum[c[n, 3 k + 1], {k, -2*n^2, 2*n^2}]
w[n_] := w[n] = Sum[c[n, 3 k + 2], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 0, z1}]   (* A211033 *)
Table[v[n], {n, 0, z1}]   (* A211034 *)
Table[w[n], {n, 0, z1}]   (* A211034 *)
LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744}, 60] (* Vincenzo Librandi, Nov 29 2016 *)

Python

from _future_ import division
def A211034(n):
    x,y,z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
    return x**2*y**2 + 2*x**2*y*z + x**2*z**2 + 2*x*y**3 + 6*x*y**2*z + 6*x*y*z**2 + 2*x*z**3 + 2*y**3*z + 2*y*z**3 # Chai Wah Wu, Nov 28 2016

Formula

From Chai Wah Wu, Nov 28 2016: (Start)

a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n > 12.

G.f.: -x*(4*x^9 + 20*x^8 + 59*x^7 + 109*x^6 + 96*x^5 + 136*x^4 + 100*x^3 + 28*x^2 + 21*x + 3)/((x - 1)^5*(x^2 + x + 1)^4).

If r = floor(n/3)+1, s = floor((n-1)/3)+1 and t = floor((n-2)/3)+1, then:

a(n) = r^2*s^2 + 2*r^2*s*t + r^2*t^2 + 2*r*s^3 + 6*r*s^2*t + 6*r*s*t^2 + 2*r*t^3 + 2*s^3*t + 2*s*t^3.

If n == 0 mod 3, then a(n) = 4*n^2*(2*n^2 + 6*n + 3)/27.

If n == 1 mod 3, then a(n) = (8*n^4 + 28*n^3 + 33*n^2 + 16*n - 4)/27.

If n == 2 mod 3, then a(n) = 8*(n + 1)^4/27. (End)